Mathematics
The Idea of Ideals
At the end of the post Unique and Non-unique Factorization, we saw how the introduction of Kummer’s “ideal numbers” remedied the lack of unique factorization in a certain ring of integers. And it was pointed out that these “ideal numbers” are just that - ideal. They have no real or actual existence. It was Richard Dedekind who gave the concept a concrete realization in terms of subsets of rings of integers known as ideals.
From Ideal Numbers to Ideals
In a series of publications in the 1870s, Dedekind expounded his theory of ideals, which has become one of the foundations of modern number theory. Taking an excerpt¹ from one of his publications, we see the line of thought that guided him in the development of the idea:
…Since a characteristic property A serves to define, not an ideal number itself, but only the divisibility of the numbers in R by the ideal number, one is naturally led to consider the set α of all numbers n of the domain R which are divisible by a particular ideal number. I now call such a system an ideal for short, so that for each particular ideal number there corresponds a particular ideal α.
Rather than work with the fanciful ideal numbers, Dedekind proposed to work with the set of all numbers they divide in the ring. From the example in the previous post, we have:
2 = β₁ x β₂
3 = β₃ x β₄
4 + √10 = β₁ x β₃
4 - √10 = β₂ x β₄
While β₁ is an ideal number and does not really exist, we know it divides 2(and consequently all multiples of 2 in the ring), it also divides 4 + √10(and all its multiples). β₁ will also divide the sum of 2 and 4 + √10, and their difference. We can make use of this divisibility relationships involving actual numbers to characterize the ideal numbers in terms of these actual numbers.
From the foregoing, the set of numbers in the ring divisible by β₁ can be written in the general form 2a + (4 + √10)b - a linear combination of 2 and
4 + √10, with a and b being elements of the ring. Let’s call the set S₁. To get a feel of the make-up of this set, let’s set b = 0, and assign some values to a; we then have S₁ = {…,-4, -2, 0, 2, 4, 6, 8, 10, 12,…}.
We applied some known divisibility properties of integers to derive the set S₁, namely;
(i) If an integer n divides another integer m, then n will also divide the multiples of m.
(ii) If an integer seperately divides two numbers a, and b, then it also divides their sum(and difference too - but that can be absorbed by the statement since difference is just negative sum; a - b= a + (-b)).
The resulting set S₁ or any such set will have the following properties; you can verify with the elements of S₁listed above:
(1) Closure under addition(or subtraction) - adding(or subtracting) any two elements in the set will give another element in the set, never outside it.
(2) Multiplying an element of the set by an element from the parent ring will give an element that is in the set.
These two properties( (1) and (2)) are what define an ideal. So formally, we can define an ideal I as a subset of a ring R having those properties.
To recap, the set of elements in the ring of integers of Q(√10) divisible by the ideal number β₁ is given by 2a + (4 + √10)b - this is the ideal generated by 2 and 4 + √10 usually written as <2, 4 + √10>, some texts write it as
(2, 4 + √10), but, to avoid confusion, we are going to stick with the angle brackets. Likewise, the ideal corresponding to β₂ is <2, 4 - √10>.
Quoting further from Dedekind:
Bearing in mind that these ideal numbers are introduced with no other goal than restoring the laws of divisibility in the numerical domain R to complete conformity with the theory of rational numbers, it is evidently necessary that the numbers actually existing in R and which are always present as factors of composite numbers, be regarded as a special case of ideal numbers. Thus if µ is a particular number of R, the system α of all numbers n= µω in the domain R divisible by µ likewise has the essential character of an ideal, and it will be called a principal ideal.
The point is, we can regard actual numbers as special cases of ideal numbers (kind of like the way we can view real numbers as complex numbers with imaginary part 0). So, an actual number like 3 for example is an ideal number with corresponding ideal;
<3> = multiples of 3 = { …-6, -3, 0, 3, 6, 9, 15, 18, 21,…}.
We can now rewrite the ideal number factorizations of 6 in Q(√10) in terms of ideals:
<2> = <2, 4 + √10> x <2, 4 - √10>
<3> = < 3, 4 + √10> x < 3, 4 - √10>
<4 + √10> = <2, 4 + √10> x < 3, 4 + √10>
<4 - √10> = <2, 4 - √10> x <2, 4 - √10>
Ideals are sets but they can be operated on like numbers, we’ll see how to do that in the next section - and then verify the above statements.
Operations on Ideals
When working with ideals it is helpful to keep in mind that they are sets. An ideal with one generator is a set consisting of all the multiples of that generator including 0. An ideal with more than one generator is a set consisting of the linear combination of the generators including 0.
Sum of Ideals
The sum of two ideals I and J is also an ideal given by:
I + J = {i + j | i ∈ I, j ∈ J }.
If I and J are infinite sets, then their sum is an infinite set consisting of all the possible results obtained by adding an element from I to an element from J. Sometimes it is possible to express this sum as an ideal with a finite set of generators.
Example: <2> + < 3> = { multiples of 2 = 2m } + { multiples of 3 = 3n }
= { numbers of form 2m + 3n } =<2, 3>.
Generally, <a> + <b> = <a, b>.
Product of Ideals
The product of two ideals I and J is the set of all finite sums of elements of the form ij with i an element of I and j an element of J. In sigma summation notation;
Now let’s verify that <2> = <2, 4 + √10> x <2, 4 - √10>
The trick here, is to find a general expression representing the elements of an ideal with more than one generator.
For some integers c, d, m, n in Q(√10), we have:
<2, 4 + √10> = { 2c + (4 + √10)d }
<2, 4 - √10> = { 2m + (4 - √10)n }. Therefore,
<2, 4 + √10> x <2, 4 - √10> = (2c + (4 + √10)d) x (2m + (4 - √10)n)
= ( 2c + 4d + d√10 ) x ( 2m + 4n - n√10 )
= ( 4cm + 8cn + 8dm + 6dn ) - √10( 2cn - 2dm )
= 2x - 2y√10 = 2(x - y√10), a multiple of 2
( with x = 2cm + 4cn + 4dm + 3dn, y = cn + dm )
Therefore <2, 4 + √10> x <2, 4 - √10> = <2>.
Principal Ideals, Non-principal ideals
We already have the definition of a principal ideal in the excerpt from Dedekind:
Thus if µ is a particular number of R, the system α of all numbers n = µω in the domain R divisible by µ likewise has the essential character of an ideal, and it will be called a principal ideal.
A principal ideal is the “system of all numbers” divisible by a particular number, in other words, the set consisting of multiples of a single element - the ideal generated by just one element of a ring. While a non-principal ideal is an ideal generated by more than one element - the linear combination of their mutiples.
With regard to unique factorization, it is important to introduce the concept of a principal ideal domain. A principal ideal domain is an integral domain (general form of a ring) where every ideal is principal. The ring of rational integers, Z, is an example of a principal ideal domain, all it’s ideals have a single generator, even if you come up with an ideal with several generators like <6, 11>, it can be shown that this non-principal ideal is equivalent to the principal ideal <1>. There is a cute, little proof of this that covers all cases, it makes use of the division algorithm.
The Division Algorithm. The division algorithm for integers states that given any two integers a and b, with b > 0, we can find integers q and r such that
0 ≤ r < b and a = bq + r. This is just the usual division of a by b giving the quotient q and remainder r, e.g. 9 = 4 x 2 + 1.
Every ideal of Z is a principal ideal - Proof
Let I be any ideal of Z, and let n be the smallest positive element of I, and m another element of I different from n, then, by the division algoritm, we can express m as m = nq + r, which gives r = m - nq, this tells us r also belong to the ideal I, since it is a linear combination of m and n which are members of I.
From the division algorithm, we know r is less than n but greater than or equal to 0. But n is supposed to be the smallest positive element of I, i.e., the next integer after 0. This means r = 0.
Setting r = 0 in m = nq + r leaves us with m = nq. That means m is a multiple of n, so every element of ideal I is a multiple of n, therefore I is a principal ideal generated by by n, I = <n>, and Z, a principal ideal domain. QED.
The significance of a ring being a principal ideal domain is that it will also be a unique factorization domain - it’s elements can be factored uniquely without necessarily introducing ideals. We’ve seen that the ring of integers in Q(√10) has some non-principal ideals, namely <2, 4 + √10>, <2, 4 - √10>,
< 3, 4 + √10>, < 3, 4 - √10>, therefore it is not a principal ideal domain. That explains why unique factorization does not hold in it.
Proper Ideals, Maximal Ideals, Prime Ideals
It is easy to see that in any ring, the ideal <0>, called the zero ideal or trivial ideal, consists of just the single element {0}, while the unit ideal <1> will contain all the elements of the ring. <0> and <1> are at two opposite extremes, any ideal in-between is called a proper ideal.
A proper ideal is an ideal that is not equal to the whole ring, it is just a subset. In the ring Z, <2>, < 3 >, < 4 > etc. are examples of proper ideals. By the way, <0> is also a proper ideal.
Still in Z, observe that the ideal <6> = {…,-18, -12, -6, 0, 6, 12, 18, 24, 30,…} is a subset of the ideal < 3 > = {…,-18,-15, -12, -9, -6, -3, 0, 3, 6, 9, 12, 15, 18,…} but < 3 > is not a subset of any ideal. Or can you work out any ideal that contains < 3 > as a subset? There is none, except the unit ideal <1> which is the whole ring. < 3 > is an example of a maximal ideal.
A maximal ideal in a ring is a proper ideal that is not a subset of any other proper ideal in the ring. Generally, if p is a prime number, then <p> will be a maximal ideal.
A fundamental property of prime numbers is that if a prime number divides a composite number then it also divides at least one factor of the composite number, prime ideals are defined after this property.
A prime ideal I in a commutative ring is a proper ideal such that if m=xy is an element of I, then at least one of x and y is also an element of I. In principal ideal domains like Z, prime ideals are generated by the prime elements of the ring, for example < 3 > is a prime ideal; 6 is an element of < 3 >, 6 = 2 x 3, and one the factors, 3 is an element of < 3 >. This condition is satisfied by all other elements of < 3 >. On the other hand, <6> is not a prime ideal; 6 = 2 x 3 is also an element of <6> but neither 2 or 3 is an element of <6>.
<0> too satisfies the condition for a prime ideal. Every maximal ideal is prime but not every prime ideal is maximal, e.g. <0> is prime but not maximal.
Reference
[1] Barnett, Janet Heine, “Richard Dedekind and the Creation of an Ideal: Early Developments in Ring Theory” (2016). Abstract Algebra. 1.
